Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

A simple Spectral Observer

Version 1 : Received: 3 March 2018 / Approved: 5 March 2018 / Online: 5 March 2018 (06:57:35 CET)
Version 2 : Received: 25 April 2018 / Approved: 26 April 2018 / Online: 26 April 2018 (07:53:32 CEST)

A peer-reviewed article of this Preprint also exists.

Torres, L.; Jiménez-Cabas, J.; Gómez-Aguilar, J.F.; Pérez-Alcazar, P. A Simple Spectral Observer. Math. Comput. Appl. 2018, 23, 23. Torres, L.; Jiménez-Cabas, J.; Gómez-Aguilar, J.F.; Pérez-Alcazar, P. A Simple Spectral Observer. Math. Comput. Appl. 2018, 23, 23.


The principal aim of a spectral observer is twofold: the reconstruction of a signal of time via state estimation and the decomposition of such a signal into the frequencies that make it up. A spectral observer can be catalogued as an online algorithm for time-frequency analysis because is a method that can compute on the fly the Fourier transform (FT) of a signal, without having the entire signal available from the start. In this regard, this paper presents a novel spectral observer with an adjustable constant gain for reconstructing a given signal by means of the recursive identification of the coefficients of a Fourier series. The reconstruction or estimation of a signal in the context of this work means to find the coefficients of a linear combination of sines a cosines that fits a signal such that it can be reproduced. The design procedure of the spectral observer is presented along with the following applications: (1) the reconstruction of a simple periodical signal, (2) the approximation of both a square and a triangular signal, (3) the edge detection in signals by using the Fourier coefficients, (4) the fitting of the historical Bitcoin market data from 2014-12-01 to 2018-01-08 and (5) the estimation of a input force acting upon a Duffing oscillator. To round out this paper, we present a detailed discussion about the results of the applications as well as a comparative analysis of the proposed spectral observer vis-à-vis the Short Time Fourier Transform (STFT), which is a well-known method for time-frequency analysis.


Signal Processing; Fourier Series; State Observer; Short Time Fourier Transform; Time-Frequency Analysis


Engineering, Control and Systems Engineering

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